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In a topological KC-space, every compact space is closed.

In a US-space, each convergent sequence has a unique limit.

So, T2 ⇒ KC ⇒ US ⇒ T1, but the converse implications do not hold.

(a): Can you give me an easy example of a US-space that is not a KC-space?

(b): Is a product of KC-spaces also a KC-space?

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    $\begingroup$ For (a), see mathoverflow.net/questions/106571/…. $\endgroup$
    – UwF
    Commented Jul 30, 2013 at 11:49
  • $\begingroup$ Are the first two lines definitions of KC and US, or are they merely properties that these types of spaces satisfy? $\endgroup$
    – André Henriques
    Commented Jul 30, 2013 at 16:07
  • $\begingroup$ the first two lines are definitions of KC and US space. $\endgroup$
    – maryam
    Commented Jul 30, 2013 at 17:15

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As pointed out in comments, the first part is answered here: A space in which sequences have unique limits but compact sets need not be closed.

The second question is answered on math.SE in the question Cartesian product of KC spaces.

I am posting this CW answer, so that this question does not remain unanswered

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